Journal of Knot Theory and Its Ramifications

We construct various functorial maps (projections) from virtual knots to classical knots. These maps are defined on diagrams of virtual knots; in terms of Gauss diagram each of them can be represented as a deletion of some chords. The construction relies upon the notion of parity. As corollaries, we prove that the minimal classical crossing number for classical knots. Such projections can be useful for lifting invariants from classical knots to virtual knots. Different maps satisfy different properties.

This paper revolves around two main results. First, we propose a "hom-set" type categorification of virtual braid groups and positive virtual braid monoids in terms of "locally" braided objects in a symmetric category (SC). This "double braiding" approach provides a rich source of representations, and offers a natural categorical interpretation for virtual racks and for the twisted Burau representation. Second, we define self-distributive (SD) structures in an arbitrary SC. SD structures are shown to produce braided objects in a SC. As for examples, we interpret the associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a (pre-)braiding. A homology theory of categorical SD structures is developed using the "braided" techniques from [Lebed, Homologies of algebraic structures via braidings and quantum shuffles, to appear in J. Algebra], generalizing rack, bar, Leibniz and other familiar complexes.

We prove that any neighborhood N of a positive twist knot , m ≥ 3 and odd with extremal Euler number can be thickened to a standard neighborhood of a maximal Thurston–Bennequin number Legendrian representative of .

Let u(K) and g(K) denote the unknotting number and the genus of a knot K, respectively. For a 3-braid knot K, we show that u(K) ≤ g(K) holds, and that if u(K) = g(K) then K is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.

For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of the Conway polynomial.

In the present paper, we will show that for any integer n > 0 there are infinitely many twisted torus knots with n-string essential tangle decompositions, and that all those knots have essential tori in the exteriors.

We prove that the size of the super-summit set of a braid can grow exponentially with the canonical length of the braid, even for pseudo-Anosov braids.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S³. In this paper, we define the Gordian complex of virtual knots by using forbidden moves. We show that for any virtual knot K₀ and for any given natural number n, there exists a family of virtual knots {K₀, K₁, …, K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the Gordian distance of virtual knots by forbidden moves d_F(K_i, K_j) = 1.

In [Towards invariants of surfaces in 4-space via classical link invariants, Trans. Amer. Math. Soc.361 (2009) 237–265], Lee defined a polynomial [[D]] for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this paper, we deal with some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial [[D]] and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial [[D]] in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of [[D]] that can be calculated from [[D]] with the help of a Gröbner basis package on computer.